GATE 2023 MA – Question 44
On ℝ let τ={U⊆ℝ:U=∅ or 1∈U}. Which statements hold?
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Correct answer: (A) First countable; (C) Separable
Explanation
At x the single set {1,x} is a neighborhood basis, proving first countability. All nonempty open sets intersect at 1, preventing Hausdorff separation of distinct points. The singleton {1} meets every nonempty open set and is dense, proving separability. Every set excluding 1 is closed, so (1,5) is already closed and does not have closure [1,5]. **A,C**.